Accounting for Fertility DeclineDuring the Transition to Growth ∗

Matthias DoepkeUCLA

October 2003

Abstract

In every developed country, the economic transition from pre-industrial stagna-tion to modern growth was accompanied by a demographic transition from high tolow fertility. Even though the overall pattern is repeated, there are large cross-countryvariations in the timing and speed of the demographic transition. What accounts forfalling fertility during the transition to growth? To answer this question, this paper de-velops a unified growth model which delivers a transition from stagnation to growth,accompanied by declining fertility. The model is used to determine whether gov-ernment policies that affect the opportunity cost of education can account for cross-country variations in fertility decline. Among the policies considered, education sub-sidies have only minor effects, while accounting for child-labor regulations is crucial.Apart from influencing fertility, the policies also have large effects on the evolution ofthe income distribution in the course of development.

JEL Classification: I20, J13, O14, O40Keywords: Growth, Fertility, Education, Child Labor, Inequality.

∗Workshop participants at Chicago, the SED annual meetings, the Max Planck Institute for Demogra-phy, the Federal Reserve Bank of Minneapolis, Wharton, Pennsylvania, Boston University, Western Ontario,UCLA, Stanford, Rochester, Virginia, UCL, Cambridge, IIES, Illinois, Minnesota, USC, UC Riverside, Duke,and Brown provided many helpful comments. I also benefited from suggestions by Sylvain Dessy, Lee Oha-nian, Daria Zakharova, and Rui Zhao. Financial support by the National Science Foundation is gratefullyacknowledged. E-mail: doepke@econ.ucla.edu. Address: UCLA, Department of Economics, 405 HilgardAve, Los Angeles, CA 90095-1477.

1 Introduction

Fertility decline is a universal feature of development. Every now industrialized countryexperienced a demographic transition form high to low fertility, accompanied by a largerise in life expectancy. Most developing countries are in the midst of their demographictransition today. Until a short time ago, economic and demographic change during de-velopment were studied in isolation: researchers studying economic growth tended toabstract from population dynamics, while demographers concentrated on explanationsfor the demographic transition. More recently, economists have started to recognize im-portant interactions between demographic and economic change, and responded by de-veloping unified models which encompass both the economic takeoff from stagnation togrowth, and the demographic transition from high to low fertility.

Accounting for fertility behavior is an important challenge for theories of development,because demographic change affects the economic performance of a country in a numberof ways. The most familiar concern is that high population growth dilutes the stock ofphysical capital, and therefore exerts a negative effect on income per capita. A perhapseven more important channel works through the accumulation of human capital. Highfertility rates tend to be associated with low education; countries with a high fertilityrate therefore accumulate less human capital. A third channel from the demographictransition to growth works through changes in the age structure of the population. Rapidfertility decline lowers the dependency ratio, since initially both the old-age and child-agecohorts are small relative to the working-age population. Hence, countries that undergo afast fertility transition experience a sizable, if temporary, boost to their level of output percapita, because the size of the labor force increases faster than the population as a whole.1

The focus of this paper is to understand why different countries undergo different demo-graphic transitions. Despite the fact that the overall pattern is repeated, the speed andtiming of fertility decline during the transition from stagnation to growth differs widelyacross countries. An example that will be used repeatedly throughout the paper is thecontrast of Brazil and South Korea. From the 1950s to the 1980s, the two countries hada very similar growth experience. Substantial growth in income per capita started at thesame time, and throughout the late 1960s and 1970s both countries were considered to be“miracle economies,” with growth rates of output per capita exceeding five percent per

1Bloom and Williamson (1998) argue that this cohort effect accounts for a large proportion of East Asia’s“economic miracle” between 1960 and 1990.

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year. Initially, fertility was similar as well, with a total fertility rate of 6.0 in both countriesin 1960. After 1960, however, fertility started to drop fast in Korea, while fertility declinewas much slower and more spread out in Brazil. Figure 1 plots the total fertility raterelative to GDP per capita. As a benchmark, data from England, the first country to expe-rience the transition from stagnation to growth, is also included in the graph. Relative toincome per capita, fertility fell much faster in Korea than it had in England, whereas thedecline is much slower in Brazil. Throughout most of the transition, for a given level ofincome per capita the fertility differential between Brazil and Korea exceeds two childrenper family.2

Why did the demographic transition proceed so much faster in Korea than in Brazil?Existing theories of growth and fertility decline do not answer this question, since theyimply that fertility is a fixed function of income per capita during the transition, so thatthe fertility transition should be identical across countries. The aim of this paper is toexplore the role of one specific explanation, namely differences in educational and child-labor policies, in explaining cross-country variations in the speed of fertility decline.3

The motivation for this explanation is twofold. First, most economic models of fertil-ity choice are built on the notion of a quantity-quality tradeoff between the number ofchildren and education per child (for example, Becker and Barro 1988). If fertility and ed-ucation are indeed joint decisions, government policies that affect the opportunity cost ofeducation should have a first-order effect on fertility. Second, we do in fact observe largevariations in educational and child-labor policies across countries during the transition togrowth. Most countries introduce education and child-labor reforms at some point dur-ing their development, but the extent and timing of these reforms varies widely. Braziland Korea are once again drastic examples. Starting in the mid-fifties (after the Koreanwar), the two countries were polar opposites in terms of their educational and child la-bor policies. In Korea, child labor was almost completely eradicated by 1960, and largeamounts of resources were devoted to building a public education system. In terms ofeducational outcomes (enrollment rates, literacy rates, average schooling) Korea was farahead of other countries at a comparable level of development. Brazil, on the other hand,spent relatively little on basic education, and was lagging far behind comparable coun-

2Similar variations in the speed of fertility decline can be observed across many other countries. Thetransition was especially rapid in the East Asian “miracle” economies such as Taiwan, Hong Kong, andSingapore. Examples of a slow transition include Malaysia, Turkey, Mexico, and Costa Rica.

3In concentrating on the role of political variables, I abstract from other factors that could also affect thefertility transition. The aim here is to derive the implications of different educational policies on the fertilitytransition in isolation; the analysis does not rule out that other factors could also matter.

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tries in terms of educational outcomes. Child labor laws were lax and loosely enforced,so that child labor was widespread well into the 1990s.4

In order to examine the role of educational policies during the fertility transition, I de-velop a growth model that generates a phase of Malthusian stagnation combined withhigh fertility, followed by a transition to a growth regime and low fertility. The model isused to explore the degree to which child-labor restrictions and education subsidies canaccount for cross-country variations in the timing and the speed of fertility decline. Thetheoretical framework consists of three key elements: An agricultural production func-tion, an industrial production function, and a quantity-quality fertility model. The mainrole of the technology side of the model, which derives from Hansen and Prescott (2002),is to generate an economic transition from stagnation to growth. The preference side ofthe model is based on the dynastic utility framework introduced by Becker and Barro(1988). Parents are altruistic and decide on the number and on the education level of theirchildren, i.e., they face a quantity-quality tradeoff. This tradeoff is essential for generatingfertility decline during the transition to growth.

To assess whether the effects of government policies are quantitatively important, themodel is calibrated to data. By simulating the calibrated model under different policyregimes, I examine how educational and child labor policies affect the fertility transition.The main finding is that these policies indeed have large effects on fertility during thetransition to growth. If parents have to pay for schooling and child labor is unrestricted,the fertility transition starts later and progresses slowly. In contrast, if education is pub-licly provided and restrictions on child labor are strict, fertility declines rapidly. Quantita-tively, education subsidies have a relatively minor impact, but accounting for child-laborrestrictions turns out to be crucial.

I also examine what happens if the same policy reforms that were instituted at the startof the take-off in Korea are introduced with a delay. This “English” regime is designed tocapture the fact that in England industrialization started before 1800, whereas the mainpolicy changes were only introduced after about 1870. In the model, with a delayed policyreform fertility stays high in the early phase of the transition, and declines rapidly oncethe reforms are introduced. This pattern is consistent with the evolution of fertility inEngland between 1800 and 1914. The timing of the education reform also has large effectson the income distribution. If the reform takes place at the start of the transition, inequal-ity stays low throughout the transition. With a delayed reform, a Kuznets curve emerges:

4See Appendix C for a description of educational policies and outcomes in Brazil and Korea.

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inequality first increases rapidly, and then decreases once the policies are changed.

This research adds to an emerging literature on long-run growth and population dy-namics. Galor and Weil (2000), Kogel and Prskawetz (2001), Jones (2001), Hansen andPrescott (2002), and Tamura (2002) all develop models that generate a transition frompre-industrial stagnation to modern growth, accompanied by a demographic transition.To the extent that fertility is endogenous, these models also have the feature that a risein the return to human capital is the driving force behind fertility decline.5 However, themodels imply fertility is a fixed function of income per capita during the transition togrowth, which is inconsistent with cross-country variations in fertility decline.

Relative to the existing literature, the main contribution of this paper is to develop a quan-titative model of the transition from stagnation to growth which can be used to under-stand cross-country differences in the transition experience.6 Apart from being interestingin its own right, addressing cross-country differences also serves as a test of the theory:if we find the true mechanism that explains the fertility decline associated with develop-ment, we should also be able to explain why the fertility decline differed so much acrosscountries. To this end, the results in this paper lend support to the quality-quantity modelof fertility choice in general.

The results also provide a new perspective on the growth-and-inequality debate. Thesame policies that influence fertility choice in the model also have large effects on theevolution of the income distribution during the transition. The main driving force be-hind the distributional effects is the endogenous fertility differential between skilled andunskilled parents. Through this differential, educational policies have large long-run ef-fects on the relative number of skilled and unskilled people, and therefore on the skillpremium.7

The rest of the paper is organized as follows. The next section introduces the model.Section 3 derives a number of theoretical properties of the model, and Section 4 discusses

5Alternative theories are based on changes in gender roles, see Galor and Weil (1996) and Lagerlof (2002),and the old-age security motive, see Morand (1999).

6Other recent studies which use quantitative theory to evaluate growth models with endogenous popu-lation include Fernandez-Villaverde (2001) on the role of capital-skill complementarity for explaining fertil-ity decline, Greenwood and Seshadri (2002) on the U.S. demographic transition, and Ngai (2000) on barriersto technology adoption.

7Fertility differentials play an important role in Moav (2001), who points to cross-country fertility dif-ferences as an explanation for persistent income inequality, and de la Croix and Doepke (2003), who linkfertility differentials within a country to economic growth. Interactions between fertility differentials andinequality are also the focus of Dahan and Tsiddon (1998), Kremer and Chen (1999), and Veloso (1999).

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the behavior of the model in the Malthusian regime, the growth regime, and the transitionbetween the two. In Section 5 I discuss the calibration procedure, and Section 6 usesthe calibrated model to assess the effect of government policies during the transition.Section 7 concludes.

2 The Model

The economy is populated by overlapping generations of people who live for two pe-riods, childhood and adulthood. Children receive education or work, do not enjoy anyutility, and do not get to decide anything. Adults can be either skilled or unskilled, de-pending on their education. In each period there is a continuum of adults of each type;NS is the measure of skilled adults, and NU is the measure of unskilled adults. Adultsdecide on their consumption, labor supply, and on the number and the education of theirchildren.

Technology

The single consumption good in this economy can be produced with two different meth-ods. There is an agricultural technology that uses skilled labor, unskilled labor, and landas inputs, and an industrial technology that only uses the two types of labor. Productionin each sector is carried out by competitive firms. The main task of the technology setupis to deliver an industrial revolution from stagnation to growth: the takeoff takes placeonce the industrial technology becomes sufficiently productive to be introduced along-side agriculture.

The firm-level industrial production function exhibits constant returns to scale, and isgiven by:

yI = AI (lS)1−α (lU )α, (1)

where yI is output (I stands for “Industry”), AI is a productivity parameter, lS and lU

are inputs of skilled and unskilled labor, and the parameter α satisfies 0 < α < 1. Sincethere are no externalities, aggregate industrial output is YI = AI (LIS)1−α (LIU )α, whereLIS and LIU are the aggregate amounts of skilled and unskilled labor employed in theindustrial sector. Since I want to abstract from bequests, there is no capital in the pro-duction function. The setup is equivalent, however, to a model with capital under the

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small-open-economy assumption, which is the approach taken by Galor and Weil (2000).8

The agricultural sector uses the two types of labor and land. The aggregate agriculturalproduction function is given by:

YF = AF (LFS)θS (LFU )θU (Z)1−θS−θU . (2)

Here YF is agricultural output (F stands for “Farm”), AF is a productivity parameter,LFS and LFU are the aggregate amounts of skilled and unskilled labor employed in theagricultural sector, and Z is the total amount of land. I assume that the parameters satisfyθS, θU > 0 and θS + θU < 1.

To abstract from land ownership and bequests, I assume that land is a public good. Fromthe perspective of a small individual firm operating the agricultural technology, there areconstant returns to labor. However, since there is a limited amount of land, labor input byone firm imposes a negative externality on all other firms. On the level of an individualfirm, for given labor inputs lS and lU of skilled and unskilled labor output yF is given by:

yF = AF (lS)θS

θS +θU (lU )θU

θS +θU , (3)

where:AF = AF [(LFS)θS (LFU )θU ]−

1−θS−θUθS+θU (Z)1−θS−θU .

Thus the total amount of labor employed has a negative effect on the productivity of anindividual firm. The specific form of the external effect was chosen such that the individ-ual production functions (3) aggregate to (2) above. As far as the analysis in this paper isconcerned, the main feature of the agricultural production function is decreasing returnsto labor. The assumption of decreasing returns is essential for generating the Malthusianregime.9 I assume that the industrial sector is more skill-intensive than the agricultural

8Specifically, if the industrial production function is y = B [(lS)1−α(lU )α]1−aka and the fixed return oncapital k is given by r, capital adjusts such that its marginal product is equal to r. Solving that condition fork gives k = (aB/r)1/(1−a)(lS)1−α(lU )α, and plugging this back into the production function gives:

y = B1

1−a (a/r)a

1−a l1−αS lαU ,

which is (1) by setting AI = B1

1−a (a/r)a

1−a . The same argument applies to the agricultural productionfunction.

9The assumption of an external effect from labor, on the other hand, is not essential, and is used only toabstract from land ownership. The model is equivalent to a setup in which all land is owned by foreignersor a separate land-owning class.

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sector:

Assumption 1 The industrial sector is more skill-intensive, i.e., the production function param-eters satisfy α < θU , which implies 1 − α > θS .

This assumption will be important for generating fertility decline during the transitionto the growth regime. The productivities of both technologies grow at constant, thoughpossibly different rates:

A′F = γF AF , A′

I = γIAI , (4)

where γF , γI > 1. The assumption of exogenous and constant productivity growth ismade to emphasize that the specific source of productivity improvements does not matterfor the qualitative results of the model; it is only necessary that productivity growth takesplace at all. Neither is it necessary to assume a change in the rate of productivity growthto explain the switch from stagnation to growth. Exogenous productivity growth allowsus to apply the model equally to countries at the technology frontier (such as England inthe 19th century) and to countries which mostly adopt existing technology (such as Koreaand Brazil in the 20th century).

The state vector x in this economy consists of the productivity levels AF and AI in theagricultural and industrial sectors, and the measures NS and NU of skilled and unskilledadults: x ≡ {AF , AI , NS, NU} The only restriction on the state vector is that it has toconsist of nonnegative numbers. Therefore the state space X for this economy is given byX ≡ R4

+.

In equilibrium, wages are a function of the state. It will be shown in Proposition 1 belowthat firms will always be operating in the agricultural sector, while the industrial sectoris only operated if the wages satisfy the condition wS(x)1−α wU (x)α ≤ AI (1 − α)1−α αα.The problem of a firm in sector j, where j ∈ {F, I}, is to maximize profits subject tothe production function, taking wages as given. Profit maximization implies that wagesequal marginal products in each sector. Writing labor demand as a function of the state,for the agricultural sector we get the following conditions:

wS(x) =AFθS

θS + θU

LFU (x)θU

LFS(x)1−θSZ1−θS−θU , (5)

wU (x) =AFθU

θS + θU

LFS(x)θS

LFU (x)1−θUZ1−θS−θU . (6)

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If the industrial sector is operating, wages have to equal marginal products as well:

wS(x) =AI (1 − α)(

LIU (x)LIS(x)

)α

if LIS(x), LIU (x) > 0, (7)

wU (x) =AI α

(LIS(x)LIU (x)

)1−α

if LIS(x), LIU (x) > 0. (8)

Instead of writing out the firms’ problem in the definition of an equilibrium below, I willimpose (5)–(8) as equilibrium conditions.

Preferences and Policies

I will now turn to the decision problem of the adults. Adults care about consumptionand the number and utility of their children. The preference structure is an extensionof Becker and Barro (1988) to the case of different types of children. The main objectiveof the preference setup is to generate a quantity-quality tradeoff between the number ofchildren and education per child. This tradeoff is essential for educational and child-laborpolicies to have an effect.

Adults discount the utility of their children, and the discount factor is decreasing in thenumber of children. In other words, the more children an adult already has, the smalleris the additional utility from another child. The utility of an adult who consumes c unitsof the consumption good and has nS skilled and nU unskilled children is given by:

cσ + β(nS + nU )−ε[nSV ′S + nUV ′

U ],

and I assume 0 < β < 1, 0 < σ < 1, and 0 < ε < 1. V ′S is the utility skilled children will

enjoy as adults, and V ′U is the utility of unskilled children, both foreseen perfectly by the

parent. The parameter σ determines the elasticity of utility with respect to consumption,β is the general level of altruism, and ε is the elasticity of altruism with respect to thenumber of children. The utilities V ′

S and V ′U are outside of the control of parents and are

therefore taken as given. The utility of children depends on the aggregate state vector inthe next period, and since there is a continuum of people, aggregates cannot be influencedby any finite number of people.

Adults are endowed with one unit of time, and they allocate their time between workingand child-raising. Children are costly, both in terms of goods and in terms of time. Raisingeach child takes ρ > 0 units of the consumption good and a fraction φ > 0 of the total

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time available to an adult. Adults also have to decide on the education of their children.Children need a skilled teacher to become skilled. It takes a fraction φS of a skilled adult’stime to teach one child. Therefore, if parents want skilled children, they have to sendtheir children to school and pay the skilled teacher. Children who do not go to schoolstay unskilled and work during childhood. Children can perform only the unskilled task,and one working child supplies φU units of unskilled labor. The parameter φU is smallerthan one since children do not work from birth on, and since they are not as productiveas adults. I also assume φU < φ, so that even after accounting for child labor there is stilla net cost associated with having unskilled children.

There are two government policies in the model, a child-labor restriction and an educa-tion subsidy. With these policies, the government can influence both components of theopportunity cost of education for a child: the value of a child’s time in terms of childlabor, and the direct schooling cost. A child-labor restriction amounts to lowering the pa-rameter φU . The government chooses a function φU (·) which determines how much timechildren work, depending on the state. Since restrictions can only lower the legal amountof child labor, I require 0 ≤ φU (x) ≤ φU for all x.10 The government also has the option ofsubsidizing a fixed amount of the schooling cost for all children at school. This expendi-ture is financed with a flat income tax, and budget balance is observed in every period.The government chooses a function δ that determines the fraction of the schooling cost tobe paid by the government, where 0 ≤ δ(x) ≤ 1 for all x. Contingent on this function, theflat tax τ is chosen to observe budget balance.

With taxes and the subsidy, the budget constraint of an adult of type i is given by:

c + ρ (nS + nU ) + (1 − δ(x)) φSwS(x) nS

≤ (1 − τ (x)) [(1 − φ(nS + nU )) wi(x) + φU (x) wU (x) nU ] . (9)

The right-hand side is after-tax income of the adult plus the income from sending un-skilled children to work. Notice that the time cost φ for each child has to be subtractedfrom the time endowment to compute labor supply. On the left-hand side are consump-tion, the goods cost for each child, and the part of the education cost for the skilled chil-dren that is paid by the parents. For simplicity, adults are not restricted to choose integernumbers of children. Also notice that there is no uncertainty in this model. Whether achild becomes skilled does not depend on chance or unobserved abilities, but is under

10In the applications below, I will consider a one-time change in child labor policy.

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full control of the parent.

In equilibrium, the wages and the utilities of skilled and unskilled people are functionsof the state vector. The maximization problem of an adult of type i, where i ∈ {S, U}, isdescribed by the following Bellman equation:

Vi(x) = maxc,nU ,nS≥0

{cσ + β(nS + nU )−ε[nSVS(x′) + nUVU (x′)]

}subject to the budget constraint (9) and the equilibrium law of motion x′ = g(x).

For the problem of a parent always to be well defined, we have to place a joint restrictionon the parameters which ensures that the effective discount factor is not higher than one.In other words, adults cannot place higher weight on the utility of their children than ontheir own utility. Since the discount factor depends on the number of children, we have toconsider the highest possible number of children, which is reached by an unskilled adultwho spends all income on children. The resulting number of children is 1/(φ − φU ).

Assumption 2 The parameters β, ε, σ, φ, φU , and γI satisfy:

β(γI)σ(

1φ − φU

)1−ε

< 1.

Here (γI)σ is the growth rate of utility along the balanced growth path which is reachedafter the switch to the industrial technology.

The fact that only parents, not children, make educational decisions leads to a marketimperfection. With perfect markets, children would be able to borrow funds to financetheir own education. In equilibrium, children would have to be indifferent between goingto school or not, so that net income of skilled and unskilled adults would be equalized.Since there are no differences in ability or stochastic income shocks, the market imper-fection is necessary to create inequality in this model. I also rule out the possibility thatparents write contracts that bind their children. Otherwise, parents could borrow fundsfrom richer adults, and have their children pay back the loan to the children of the lender.

It will be shown in Section 3 below that the adults’ problem has only corner solutions.Adults either send all their children to school, or none of them; there are never bothskilled and unskilled children within the same family. It is possible, however, that adultsof a specific type are just indifferent between sending all their children to school or none.

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In that case, some parents of a given type might decide to have skilled children, while oth-ers go for the unskilled variety. In equilibrium, the typical situation will be that all skilledparents have skilled children, while there are both unskilled parents with unskilled chil-dren and unskilled parents who send their children to school. In other words, there isupward intergenerational mobility.

In the definition of an equilibrium I have to keep track of the fractions of adults of eachtype who have skilled and unskilled children. The function λi→j(·) gives the fraction ofadults of type i who have children of type j, as a function of the state x. For each type ofparent and for all x ∈ X these fractions have to sum to one:

λS→S(x) + λS→U (x) = λU→S(x) + λU→U (x) = 1. (10)

The policy function nj(i, ·) gives the number of children for i-type parents who have j-type children as a function of the state.

The remaining equilibrium conditions are the labor-market clearing constraints, the bud-get constraint of the government, and the laws of motion for the number of skilled andunskilled adults. These constraints and a formal definition of an equilibrium are given inAppendix A.

3 Analytical Results

This section derives a number of theoretical results that will be useful for describing theequilibrium behavior of the model. First, I analyze the two production sectors, and thenI turn to the decision problem of an adult in the economy. All proofs are contained in theappendix.

On the technology side the main result is that while the agricultural sector is always oper-ating, industrial firms produce only if industrial productivity is sufficiently high relativeto wages. The following propositions derive the conditions that are necessary for produc-tion in industry and agriculture.

Proposition 1 Firms will be operating in the industrial sector only if the skilled and unskilledwages wS(x) and wU (x) satisfy the condition:

wS(x)1−α wU (x)α ≤ AI (1 − α)1−α αα. (11)

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In the agricultural sector, firms will be operating given any wages.

It is easy to check whether the industrial sector will be operated for a given supply ofskilled and unskilled labor. We can use conditions (5) and (6) to compute wages in agri-culture under the assumption that there is agricultural production only. If the resultingwages satisfy condition (11), the industrial technology is used. Skilled and unskilled la-bor is allocated such that the wage for each skill is equalized across the two sectors. Ifcondition (11) is violated, production takes place in agriculture only.

In equilibrium, initially only the agricultural technology is used. Given that there is pos-itive productivity growth in industry, at some point the industrial technology becomessufficiently productive to be introduced alongside agriculture, and an “industrial revolu-tion” occurs. This behavior arises from an interaction between the properties of the twoproduction sectors and the population dynamics in the model. Since population is de-termined by fertility decisions, I will now turn to the decision problem of an adult in themodel economy.

From the point of view of an adult, the utility of a potential skilled or unskilled child isgiven by a number that cannot be influenced. There are no individual state variables,and the utility of children is determined by fertility decisions in the aggregate, whichadults take as given since there is a continuum of people. This allows us to analyzethe decision problem of an adult without solving for a complete equilibrium first. Inthis section, we will analyze the decision problem of an adult who receives (after-tax)wage w > 0 and who knows that skilled children will receive utility VS > 0 in the nextperiod, whereas unskilled children can expect VU > 0. I restrict attention to positiveutilities, because if children receive zero utility, it is optimal not to have any children. Inorder to keep notation simple, I will express the cost of children directly in terms of theconsumption good. The cost for a skilled child is pS, and the cost for an unskilled childis denoted as pU . For example, without government policies we have pS = φw + φSwS + ρ

and pU = φw − φUwU + ρ. We always have pS > pU ; skilled children are more expensivethan unskilled children.

We can now write the maximization problem of an adult as:

maxnS ,nU≥0

{(w − pSnS − pUnU )σ + β(nS + nU )−ε[nSVS + nUVU ]

}. (12)

An alternative way of formulating this problem is to imagine the adults as choosing thetotal education cost E they spend on raising children and the fraction f of this cost that

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they spend on skilled children. The number of children is then given by nS = fE/pS andnU = (1 − f )E/pU . In this equivalent formulation, the maximization problem of the adultis:

max0≤E≤w,0≤f≤1

{(w − E)σ + βE1−ε(f/pS + (1 − f )/pU )−ε[fVS/pS + (1 − f )VU/pU ]

}. (13)

The first result is that the decision problem of an adult has only corner solutions:

Proposition 2 For any pair {E, f} that attains the maximum in (13) we have either f = 0 orf = 1.

Adults choose either skilled or unskilled children, but never mix both types in one family.We can gain intuition for this result by considering the model with ε = 0 (which is ruledout by assumption), in which case both the utility gained from having children and thecost of children are linear in the numbers of the two types of children. If we have VS/pS =VU/pU , the adult is indifferent between unskilled and skilled children and any convexcombination of the two types. However, if we now have ε > 0, as assumed, the term(f/pS + (1 − f )/pU )−ε in (13) becomes a convex function of f , and the adult will choose acorner solution.

Given that there are only corner solutions, the optimal number of children can be de-termined by separately computing the optimal choices assuming that there are only un-skilled or only skilled children. We can then compare which type yields higher utility.Parents who decide to have children of type i solve:

max0≤ni≤w/pi

{[(w − pini)]σ + β(ni)1−εVi

}.

The first-order condition can be written as:

β(1 − ε)(w − pini)1−σVi = σpi(ni)ε. (14)

There is a unique positive ni solving this equation, and the second-order condition for amaximum is satisfied given that we assume 0 < σ < 1 and 0 < ε < 1. The optimal numberof children ni is increasing in the children’s utility Vi and in the wage w. Thus children area normal good in this model. On the other hand, if the cost of children pi is proportionalto the wage w, the optimal number of children decreases with the wage. Thus if the costof children is a pure time cost, the substitution effect outweighs the income effect.

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If an adult is indifferent between skilled and unskilled children, the total expenditure onchildren does not depend on the type of the children.

Proposition 3 An adult is indifferent between skilled and unskilled children if and only if thecosts and utilities of children satisfy:

VS

(pS)1−ε=

VU

(pU )1−ε. (15)

If an adult is indifferent, the total expenditure on children does not depend on the type of childrenthat is chosen.

Propositions 2 and 3 have implications for intergenerational mobility in the model. Propo-sition 3 states that for given utilities of skilled and unskilled children the ratio of theprices of skilled and unskilled children determines whether parents send their childrento school. As long as the wage for skilled labor is higher than the unskilled wage, skilledchildren are relatively cheaper for skilled parents, since wS > wU implies:

φwS + φSwS + ρ

φwS − φUwU + ρ<

φwU + φSwS + ρ

φwU − φUwU + ρ.

The term on the left-hand side is the ratio of the prices for skilled and unskilled childrenfor skilled adults, and the right-hand side is the ratio for unskilled adults. The cost oftime is higher for skilled adults, because the skilled wage is higher than the unskilledwage. Since the opportunity cost of child rearing makes up a larger fraction of the cost ofunskilled children, unskilled children are relatively more expensive for skilled parents.

Since the relative price of skilled and unskilled children differs for skilled and unskilledparents, it cannot be the case that both types of adults are indifferent between the twotypes of children at the same time. Since skilled children are relatively cheaper for skilledparents, in equilibrium there are always skilled parents who have skilled children. Oth-erwise, there would be no skilled children at all, which cannot happen in equilibrium.Likewise, there are always unskilled adults with unskilled children.

Taking these facts together, exactly three situations can arise in any given period. The firstpossibility is that skilled parents strictly prefer skilled children, while unskilled parentsstrictly prefer unskilled children. In that case, there is no intergenerational mobility. Thesecond possibility is that skilled parents are indifferent between the two types of children,while all unskilled parents have unskilled children. The third option is that all skilled

14

parents have skilled children, while the unskilled adults are indifferent between the twotypes. This last case is the typical one along an equilibrium path, as will be explainedin more detail later. In this situation, there is upward intergenerational mobility, becausesome unskilled adults have skilled children, but no downward mobility.

The following proposition sums up the implications of these results for an equilibrium.

Proposition 4 In equilibrium, for any x ∈ X such that wS(x) > wU (x), the following must betrue:

• A positive fraction of skilled adults has skilled children, and a positive fraction of unskilledadults has unskilled children:

λS→S(x), λU→U (x) > 0.

• Just one type of adult can be indifferent between the two types of children:

λS→U (x) > 0 implies λU→S(x) = 0,

λU→S(x) > 0 implies λS→U (x) = 0.

• Specifically, λS→U (x) > 0 implies:

(φwS(x) + φSwS(x) + ρ

φwS(x) − φUwU (x) + ρ

)1−ε

=VS(g(x))VU (g(x))

,

and λU→S(x) > 0 implies:

(φwU (x) + φSwS(x) + ρ

φwU (x) − φUwU (x) + ρ

)1−ε

=VS(g(x))VU (g(x))

.

4 Outline of the Behavior of the Model

Assuming that the economy starts at a time when productivity in industry is low com-pared to agriculture, the economy evolves through three different regimes: The Malthu-sian regime, the transition, and the growth regime. In the Malthusian regime the indus-trial technology is too inefficient to be used for some time. Therefore the model behaves

15

like one in which there is an agricultural sector only. The economy displays Malthusianfeatures—wages stagnate, and population growth offsets any improvements in produc-tivity.

There are three key features of the model that generate the Malthusian regime. First,it is essential that children are a normal good. This property ensures that populationgrowth increases once improvements in technology lead to higher wages. The secondnecessary assumption is that the agricultural technology exhibits decreasing returns tothe size of the labor force. This feature ensures that higher population growth depresseswages, which pushes the economy back to the steady state. The third key assumptionis that there is a goods cost ρ for each child. Without this cost, it would be possiblethat population growth stays ahead of productivity growth, and wages converge to zero.Taken together, these features create a Malthusian feedback in which rising wages raisepopulation growth, but population growth depresses wages, resulting in a steady state inwhich wages are constant.

The values for all variables in the Malthusian regime can be determined by solving a sys-tem of steady-state equations under the assumption that only the agricultural technologyis used. For a wide range of parameter choices, there is a solution to the steady-state equa-tions in which both wages and the ratio of skilled to unskilled adults are constant11. In thesteady state, average fertility is higher for unskilled adults. While all skilled adults sendtheir children to school, there are some unskilled parents with unskilled children andothers with skilled children. Wages depend on preference parameters and the growthrate of agricultural productivity, but are independent of the level of productivity, sinceproductivity growth is exactly offset by population growth.

Even though only the agricultural production function is used in the Malthusian regime,the productivity of the industrial technology is increasing over time. At some point, pro-ductivity in industry reaches a level at which industrial production is profitable at thewages that prevail in the Malthusian regime. From that time on the industrial technol-ogy will be used alongside the agricultural technology. Since population growth does notdepress wages in industry, wages and income per capita start to grow with productivityin the industrial sector. While the model assumes that the growth rate of productivity inindustry is constant even before the technology is used, all that is necessary is that pro-

11If the schooling cost is very high, it is possible that no solution exists and the fraction of skilled adultsconverges to zero. Also, if productivity growth and the cost of children are very high it is possible thatpopulation growth is not high enough to offset productivity growth. Neither case arises in the calibratedmodel.

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ductivity growth in industry is bounded away from zero. This is a natural assumptionsince technology in industry and agriculture is complementary, in the sense that inven-tions which are useful for agriculture also have industrial uses. For example, James Watt’scontributions to the design of steam engines would not have been possible without priordiscoveries in mechanics and metallurgy that were originally aimed at improving agri-cultural (and perhaps military) technology.

The evolution of fertility and the income distribution once the transition starts dependson the specific properties of the industrial production function. Given Assumption 1 (pro-duction in industry is more skill-intensive than production in agriculture), the introduc-tion of the industrial technology increases the wage premium for skilled labor. This in-creases the returns to education, so that more unskilled adults will choose to have skilledchildren, resulting in higher social mobility. The overall effect on fertility is uncertain.The increased demand for expensive skilled children would tend to lower fertility, butthen since wages start to grow, the utility of children relative to their parents increases,which tends to increase fertility. The transition can be influenced by public policy. Both aneducation subsidy and child-labor restrictions lower the relative cost of skilled children,so that both policies have a positive effect on the number of children going to school.The effects on fertility, however, are different. Since a subsidy lowers the cost of children,an education subsidy tends to increase fertility, even though more children are going toschool. Child labor restrictions, on the other hand, increase the cost of children, andtherefore lead to lower fertility. The quantitative importance of public policies during thetransition will be assessed below.

If productivity growth in industry is sufficiently high, the fraction of output producedin industry will increase over time, until the agricultural sector ultimately becomes neg-ligible. The economy will then reach a balanced growth path, the growth regime. Herethe model behaves like one in which there is the industrial technology only. All variablescan be computed by solving a system of balanced-growth equations. Whether fertility ishigher in the growth regime than in the Malthusian regime is determined by the relativeimportance of skill in the two technologies. If the industrial technology is sufficientlyskill-intensive, in the growth regime most children will go to school. Since schoolingis costly, this will tend to lower fertility and population growth. On the other hand, aswages grow, the physical cost ρ of children ultimately becomes negligible. This effectmakes children relatively cheaper in the growth regime, which will tend to increase fertil-ity. Unless the schooling cost is very high, the ratio of skilled to unskilled adults reaches

17

a fixed number in the growth regime.12 Population growth and fertility are constant,and wages and consumption grow at the rate of technical progress. Average fertility islower for skilled than for unskilled adults. This would be true even if the schooling costwere zero and if there were no child labor, since then the only remaining cost of childrenwould be a time cost. It was shown in Section 3 that if there is a pure time cost for havingchildren, wages and fertility are negatively related. With positive schooling cost and childlabor the relative cost of skilled children increases, and since relatively more skilled adultshave skilled children, this will further increase the fertility differential between the twotypes of adults. Given that fertility is higher for unskilled adults in the balanced growthpath, it has to be the case that some unskilled adults have skilled children. Otherwise thefraction of unskilled adults would increase over time. Therefore unskilled adults are justindifferent between the two types of children.

In the limit-economy without agriculture wages are determined by the ratio of skilled tounskilled labor supply. The only required state variables are therefore the ratio of skilledand unskilled adults and the productivity level in industry. The setup can further besimplified by noting that the period utility function is of the constant-elasticity form, andthat wages are linear in the productivity level. This results in value functions that arehomogeneous in the industrial productivity level:

Vi(AI , NS/NU ) = AσI Vi(NS/NU ).

This reduces the growth regime essentially to a one-dimensional system, with the ratio ofskilled to unskilled adults as the state variable.

I will now turn to the question whether public policy can have large effects during thetransition. In the following sections, I calibrate the model parameters, and simulate themodel under different assumptions on policies.

5 Calibration

In this section I describe the procedure for calibrating the model parameters. Since Iuse the model to examine whether policy differences across countries can account fordifferent transition experiences, it would be counterproductive to choose parameters to

12If the schooling cost is too high, the number of skilled adults converges to zero over time.

18

closely match data from one specific country. Rather, I choose parameters such that in thegrowth regime the model matches certain features of modern industrialized countries,while in the Malthusian regime the model resembles a pre-modern economy. For thegrowth regime, I use current U.S. data, while for the pre-modern economy I rely mostlyon England, where the available data is of relatively high quality. Since the calibrationprocedure pins down only the two ends of the time line, we can use the transition periodfor testing the model.

I start by describing the parameter choices that are determined by features of the growthregime. The parameter γI , the rate of technological progress in the industrial sector, de-termines the growth rate of per-capita output in the growth regime. In the United States,real GDP per capita increased on average by 1.9% per year in the period from 1960 to1992. I therefore chose the yearly growth rate of productivity in the industrial sector to be2%. Since a model period is 25 years, this gives a value for γI of 1.64.

Given that there are strict compulsory schooling laws and child-labor restrictions in theU.S., the parameters for the growth regime are calibrated under the assumption that childlabor is ruled out. The schooling cost parameter φS determines the number of teachersand the student-teacher ratio in the growth regime. According to the Digest of Educa-tion Statistics (U.S. Department of Education, 1998), in the U.S. teachers on all levels ofeducation make up 1.5% of the American population, and there are about 16 students perteacher. Since, unlike in the model, in the real world each teacher teaches more than onegeneration of students, we cannot match both values at the same time. If the ratio of teach-ers to population is matched, class sizes would be too big, and if class size is matched,there would be too many teachers. As a compromise, I chose φS to be 0.04, which resultsin a class size of 21, and 1.7% of the population are teachers.13 The time cost φ was thenchosen to match the total expenditure on children in the United States, estimated for 1992by Haveman and Wolfe (1995). According to their estimates, parental per-child expendi-tures were $9,200 or about 38% of per capita GDP. I chose φ to be 0.155, which leads tothe same ratio of per-child expenditures to GDP per capita. Knowles (1999) calibrates thesame parameter to other estimates of the cost of children and arrives at a similar value of0.15.

Using data from 1975, Jones (1982) finds that in Britain the difference in total fertilitybetween women with elementary education or less and women with secondary or highereducation is about 0.4, and the corresponding value for the U.S. is about 0.5. I chose the

13Variations of the model which match either statistic exactly lead to essentially the same outcomes.

19

preference parameters σ, ε and β to be consistent with a fertility differential of 0.5 betweenskilled parents and unskilled parents who have unskilled children, and a total fertility rateof 2.0, which matches current fertility in the United States. The choices σ = 0.5, ε = 0.5,and β = 0.132 are consistent with these observations.

According the Digest of Education Statistics, in 1994 total expenditures on educationwere 7.3% of GDP, while public expenditures were 4.8%, or roughly two thirds of thetotal. These numbers exclude expenditures by parents and students, like textbooks andtransportation. The government therefore pays for less than two thirds of all educationalexpenditures. In the model, I chose the fraction δ of education cost paid for by the gov-ernment to be 0.5.

The technology parameter α, the share of unskilled labor in the industrial productionfunction, mainly determines the ratio of skilled to unskilled people in the growth regime.It is hard to match this ratio to data, since there are more than two skill levels in the realworld. If we define skill to mean completed high school, skilled people would make upmost of the population, since already today almost 90% of recent school cohorts satisfythis criterion. On the other hand, if skilled means completed college education, the num-ber of skilled people would drop below 30%. Since college education was rare in someof the countries and time periods I am interested in, I chose a compromise with higherweight on high school. The parameter α was chosen to be .22, which results in 75% of thepopulation in the growth regime to be skilled.

I now turn to the Malthusian regime. Most parameters are identical to the growth regime,we only need to calibrate the agricultural technology and the child-labor parameter. Theparameter γF , the rate of technological progress in the agricultural sector, determinesfertility in the Malthusian regime. In Britain the total fertility rate was about 4.0 in 1700,and values in other European countries were similar. The value γF = 1.32 yields thisfertility rate in the Malthusian regime. The fixed cost for children ρ is a scale parameterand can be chosen arbitrarily. I chose ρ = .001.

The child-labor parameter φU is hard to calibrate, since there is only limited evidence onthe extent of child labor before the industrial revolution. However, φU has to be chosensufficiently large to allow a Malthusian regime. If φU is small, population growth doesnot catch up with technological progress, and per capita output grows already beforethe introduction of the industrial technology. On the other hand, if φU is too large, thenumber of skilled agents converges to zero after the industrial technology is introduced.My choice of φU = .07 is in the middle ground and is consistent both with Malthusian

20

stagnation and a growth regime with a positive fraction of skilled agents. Adjusting forlabor supply, this implies that nine children are equivalent to one unskilled worker. Thebehavior of the model is sensitive to the choice of φU ; by making φU large, we can increasethe effects of eliminating child labor. It is therefore important to ascertain that φU is notunrealistically large. From the perspective of the parent, the key question is how much ofhousehold income is contributed by children. Horrell and Humphries (1995) find that ina sample of working-class families in England around 1800 children contributed between27% and 32% of household income, whereas in the model income from child labor makesup only 25% of income in families with working children.

For the parameters of the agricultural technology, I chose θS = .1 and θU = .5. Withthese choices, the share of output going to land rents is 40%, and skilled adults make upabout 5% of the population in the Malthusian regime. A lower bound on the land sharecan be derived from the English National Accounts for 1688 in Deane and Cole (1969).At that time rents made up 27% of national income. Since some of the 34% of nationalincome going to non-rent property income and profits are also derived from land, 30%is a reasonable lower bound for the share of land. On the other hand, share-croppingcontracts typically allocated 50% of output to the land owner. Since not all of nationalincome was derived from agriculture, 50% is then an upper bound for the land share. Mychoice lies in the middle between these bounds.

6 Computational Experiments

This section uses the calibrated model to determine whether policy changes can havelarge effects during the transition. I simulate the model under three different assump-tions on government policies, roughly corresponding to the cases of Brazil, Korea andEngland mentioned earlier.14 In the stylized Brazilian policy, there is no policy reform.Parents pay for education, and child labor is unrestricted. In the stylized Korean pol-icy, the economy starts out with the same policies. Once the economy starts to leave theMalthusian regime, however, there is an immediate switch to the policies that were thebasis for calibrating the growth regime. Under these policies, the government pays for50% of the education cost for skilled children, and child labor is ruled out. While in the

14See Appendix C for a description of actual policies in these countries. The policies used in the sim-ulations are not intended to be exact copies of real-world policies. Rather, I use stylized, one-time policychanges to assess the role of educational policies in general. Especially for Brazil, the policy description inthe model is not “fair” for more recent years: support for education has increased in the last decades (andfertility has been falling faster).

21

Korean policy the reform takes place immediately once the industrial technology is in-troduced, in the stylized English policy the same reforms are carried out with a delay ofthree periods (or 75 years) after the beginning of the transition. All economies start fromidentical conditions. The initial values for agricultural productivity and the number ofskilled and unskilled adults are chosen to start the economies in the Malthusian steadystate. The productivity in industry is chosen such that the transition starts two periodsafter the start of the simulation. Apart from the different timing of policy reforms (noreform under the Brazilian policy, immediate reform under the Korean policy, delayedreform under the English policy), all simulations are identical.

The outcomes can be summarized as follows. The timing of policy reforms has a ma-jor impact on the timing and speed of the fertility transition. Without a policy reform,the decrease in fertility during the transition to growth is small. When a reform is intro-duced, it causes an immediate drop in fertility, and the transition to replacement fertilityis completed within two generations after the reform. In addition, lowering the oppor-tunity cost of education increases growth in GDP per capita, since the fraction of skilledworkers in the population increases. Out of the two policies considered, child-labor re-strictions account for the bulk of the results, while the effects of education subsidies arecomparatively small. The same policy reforms also have a major impact on the evolutionof the income distribution. As long as no policies are introduced, inequality rises duringthe transition, since the skill premium goes up. Once the policy reform lowers the op-portunity cost of education, however, the fraction of skilled workers increases over time,which lowers the skill premium and overall inequality.

Figure 2 shows the simulated path for GDP per capita and the total fertility rate underthe different policies. In the graphs for the Korean and English policies, the Brazilian out-comes are included (with dashed lines) as the benchmark case to facilitate comparisons.The time axis is labeled such that each model period corresponds to 25 years, and thestart of the transition to the growth regime is placed in the year 1950 for the Brazilian andKorean policies, and in 1800 for the English policy. GDP per capita is initially constant,since the simulation starts in the Malthusian regime where wages are constant. Once theindustrial technology is introduced, incomes start to grow.

The total fertility rate increases slightly before the start of the transition, and then declinesin all simulations. The speed and timing of fertility decline, however, varies across thesimulations. With no reform, fertility decreases only slowly to a total fertility rate of about3. With an immediate reform, the total fertility rate drops below 3 already at the start of

22

the transition, and replacement fertility is reached in less than two generations after thetake-off. The maximum difference in fertility across the Brazilian and Korean simulationin any year is about 1.5, or about 60% of the maximum difference between Korean andBrazilian data in Figure 1. Thus, while the model does not account for all of the observeddifferential, the effect of the policies is substantial. With the delayed policy reform of theEnglish simulation, fertility decline starts right after the beginning of the transition aswell, but speeds up after the reforms are carried out (1875 in the simulation). Relative tothe actual English experience, the model with a delayed education reform does a good jobin accounting for the rapid fertility decline around the end of the 19th century. The maindeviation between model and data is that the model not reproduce the initial increase infertility between 1700 and 1820 in England, as well as the peak fertility level of about 6.0in Korea and Brazil. Across all policy options, the maximum fertility level reached in thesimulations is 4.8.

Since fertility and education decisions interact, the same policies that affect the speed offertility decline also influence economic growth in the model. The highest growth rateis achieved under the Korean policy of an immediate education reform. Under the Ko-rean policy, the average growth rate of GDP per capita between 1950 and 2000 is 4.2%,which compares to a growth rate of only 3.3% under the Brazilian Policy. In the data,we also observe that Korea, as well as other countries with a fast demographic transition,experienced faster growth in GDP per capita than slow-transition countries like Brazil.According to the Penn World Tables, the average growth rate in GDP per capita between1960 and 1990 was 6.7% in Korea, but only 2.8% in Brazil. Even if we consider the shorterperiod from 1960 to 1980 (which excludes the Latin American debt crisis in the 1980s),average growth was 6.2% in Korea versus 4.4% in Brazil. The growth rate differential inthe data is larger than the differential in the model, which suggests that social policies andeducation decisions are not the only factor explaining the relative performance of Braziland Korea. Given that the two countries are different along a number of dimensions, itwould in fact be very surprising if the policies considered here were the main explana-tion for differences in economic performance. Nevertheless, the results suggest that thepolicies and their effects on fertility and education decisions are an important part of theoverall explanation.

The reason for the growth-rate differential in the model is is that with education subsidiesand child-labor restrictions, the skilled fraction of the population increases fast, which

23

raises output since the industrial technology is skill-intensive.15 Under the Korean policy,the fraction of skilled adults in the population increases from 5% in 1950 to almost 70% in2000. Under the Brazilian policy, the increase is from 5% to only about 33%. In the data,we observe large differences in average skill between Korea and Brazil as well. In Korea,average years of schooling of the population aged 25 and over increased from 3.2 in 1960to 7.8 in 1985. In Brazil, the increase was from 2.6 in 1960 to only 3.5 in 1985. Notice that ofthe three channels through which the speed of fertility decline affects GDP per capita onlythe human-capital channel is present in the model. This channel captures that becauseof the quantity-quality tradeoff, parents increase the education of their children whenfertility falls. The model abstracts from the capital-dilution channel (lower populationgrowth increases the capital stock per worker) and the cohort-size channel (rapid fertilitydecline lowers the dependency ratio). If we extended the model to capture these effectsas well, the predicted growth differential would be even greater.

Apart from influencing fertility and output, policy changes also affect the income distri-bution. The graphs on the left-hand side of Figure 3 shows income Gini coefficients forthe simulations. In the model, inequality is determined by the skill premium, which inturn depends on the relative supply of skilled and unskilled labor. The Korean policyincreases the relative number of skilled adults, which offsets the increased demand forskilled labor in the industrial sector. Therefore, in the simulation with an immediate pol-icy reform inequality is generally low and decreases slightly during the transition. Theinitial increase in the Gini at the time of the policy reform occurs since the demand for ed-ucation suddenly increases. This temporarily raises the skill premium, because teachersare skilled. In the Brazilian simulation unskilled adults continue to make up the majorityof the population. The increased demand for skill in the industry translates into an in-creased wage premium, so that inequality increases during the transition. With a delayedpolicy reform, inequality first increases, then decreases during the transition.

Thus only with the English policy do we observe the pronounced inverse-U-shaped rela-tionship between income and inequality known as the Kuznets curve. In the real world,this pattern of inequality over the course of development was observed in the countriesthat industrialized first (Kuznets (1955) looked at England, Germany, and the UnitedStates), but for countries that started to grow more recently the evidence is mixed. Specif-ically, neither Korea nor Brazil exhibit a Kuznets curve. In Korea inequality was generally

15Because productivity growth is exogenous, the effect on growth rates is transitory, but there is a perma-nent level effect as long as policies differ.

24

low, and there was little change in the income distribution over time. Between 1955 and1985, the Gini coefficient varies between .3 and .4. In Brazil inequality was always high,with a Gini between .5 and .6. The income share going to the poorest 20% of the Koreanpopulation is about the same as the share going to the poorest 40% in Brazil. Even consid-ering that the model abstracts from some features which are relevant for explaining theincome distribution16, the results do show that interactions of public policies with fertilitydecisions can have large effects on the growth-inequality relationship. This may providean explanation for why Kuznets curves show up in some countries, but not in others.

A more direct test of the distributional implications of the model would look at wageinequality, and especially the premium for skilled labor, since the skill premium is thedriving force behind changes in inequality in the model. The skill premium reacts to thepolicies through the the general-equilibrium effect of education and fertility decisions onthe relative supply of skilled and unskilled labor. Clearly, this general equilibrium effectis central to the results: fertility falls fast in the Korean simulation precisely because thepolicies deliver a low skill premium, which means that education is affordable even forunskilled parents. Park, Ross, and Sabot (1996) compare wage inequality in Brazil and Ko-rea in 1976 and 1986. Consistent with the predictions of the model, they find much higherskill premia in Brazil than in Korea. Moreover, there is a substantial decline in wage in-equality in Korea between 1976 and 1986. Specifically, Park, Ross, and Sabot estimate astandard earnings equation, i.e., the logarithm of earnings is regressed on education andwork experience. In 1976, high school graduates in Brazil earned a 60 percent premiumrelative to workers who finished primary school, while the premium was only 35 percentin Korea. For university graduates, the premium (relative to primary school) was 201percent in Brazil versus 121 percent in Korea. In 1986, the differences were even morepronounced. In Brazil, the premia for high school and university graduates were still 55and 188 percent, respectively, while in Korea the premia had fallen to just 10 percent and76 percent.

In the model, changes in overall fertility are driven by differences in fertility betweenskilled and unskilled parents. The graphs on the right-hand side of Figure 3 break downthe total fertility rate in each simulation by the type of the parent. In the Brazilian simu-lation there are no education subsidies or child-labor restrictions. Unskilled children are

16For the comparison of Brazil and Korea, the role of land ownership is essential. After a land reform atthe end of Japanese occupation, land ownership was dispersed in Korea, but always highly concentratedin Brazil. Indeed, if concentrated land ownership is introduced, the model generates high inequality withalmost no time trend under the Brazilian policy.

25

cheap relative to skilled children, and therefore unskilled parents continue to have manychildren throughout the transition. Initially, average fertility increases for both groupsof parents, and the aggregate total fertility rate falls only because skilled adults begin tomake up a larger fraction of the population. The fertility differential between skilled andunskilled parents stays large. Under the Korean policy, the fertility differential betweenthe two types of parents declines rapidly after the policy change. The abolition of child la-bor and education subsidies imply that skilled children are only slightly more expensivethan unskilled children, which results is a small fertility differential between skilled andunskilled parents. The same effect occurs in the English simulation after the new policiesare introduced.

In the data, in line with the predictions, the available evidence shows small fertility dif-ferentials in Korea and large differentials in Brazil. According to United Nations (1995),in 1986 Brazilian women without formal education had a total fertility rate of 6.7, whilefor women with seven or more years of education the number is 2.4. Thus the fertilitydifferential amounts to more than four children. Alam and Casterline (1984) report thatin 1974 the total fertility rate for Korean women without formal education was 5.7, whilefor women with seven or more years of education the rate is 3.4. This gives a fertilitydifferential of 2.3, roughly half of the differential in Brazil.17 England, as well as otherindustrialized countries, has small fertility differentials within the population today incomparison to developing countries.

In summary, we see that the policies considered have large effects on fertility decline dur-ing the transition to growth. These differences translate into further implications for GDPper capita, average education levels, fertility differentials, and the income distributionwhich line up well with the evidence for our examples Brazil and Korea. So far the analy-sis of public policy was restricted to a combination of two policies, an education subsidyand child-labor restrictions. A natural question to ask is which of the two policies is moreimportant for generating the results described above. It turns out that the child-labor re-striction has a much bigger impact than the education subsidy. Figure 4 shows outcomesfor GDP per capita and fertility if either education subsidies or child labor restrictionsare introduced in isolation. If only a child-labor restriction is introduced, the evolutionof income per capita and fertility is similar to the Korean simulation, which introducesboth policies. With a child-labor restriction there is an unambiguous increase in the cost

17Unfortunately, there is no data for Brazil and Korea for the same year. However, given the overall fallin fertility, it seems likely that the fertility differential by education in Korea was even lower in 1986 than in1974.

26

of unskilled children, which reduces fertility even faster compared to the case when bothpolicies are combined. In contrast, when there is only an education subsidy, fertility ac-tually increases initially, and the subsequent decline is much smaller than in the Koreansimulation. The reason is that the education subsidy works in two different ways. Sinceskilled children become cheaper, more unskilled parents decide to educate their children,which lowers fertility. At the same time, however, fertility increases for the skilled par-ents who would have decided to educate their children even without the subsidy. Thetwo effects are partially offsetting, and the overall impact is small. These results under-line the importance of including the cost of the children’s time in the opportunity costof education. In fact, if we did not account for the cost of the children’s time, it wouldbe rather surprising that in England fertility fell fastest just at the time when free publiceducation was introduced.

A welfare analysis of the policies is made difficult by the fact that skilled and unskilledpeople have opposing interests. Therefore the policies cannot be ranked using the Paretocriterion. The education subsidy, however, is beneficial for adults of both skills in theperiod when the subsidy is introduced. Future generations of skilled people have lowerutility with the policy because of a lower skill premium. A child-labor restriction alonehurts both skills in the period when it is introduced, since the unskilled parents lose in-come, and the utility of skilled children falls. However, future generations of unskilledpeople benefit from child-labor restrictions. A combination of the two policies yields thehighest growth path for GDP per capita, but future skilled generations are still better offwithout the policies.

7 Conclusions

This paper develops a theory that is consistent with a phase of stagnation during whichthe economy exhibits Malthusian features, followed by a transition to a growth regime.The economic transition to growth is accompanied by a demographic transition from highto low fertility. The model is used to assess whether child-labor restrictions and educationsubsidies can have large impacts on the fertility decline that accompanies the transitionto growth.

The possibility of child labor implies that the value of a child’s time becomes part of the

27

opportunity cost of education.18 Without accounting for the value of the children’s time,it would be very puzzling that in many countries fertility fell fastest just at the time whenfree public education was introduced. When there is no alternative use for a child’s time,free education lowers the cost of children and therefore should lead to higher fertility. Incontrast, once we account for the full opportunity cost of education including potentialchild labor income, educating a child is costly even if schools are free. A steep decline infertility right after major education reforms is therefore just what we should expect to see.

Indeed, we find that education and child labor policies have large effects on the fertilitytransition. The policies also determine the evolution of income per capita, average edu-cation, and the income distribution of a country during the transition to growth. Whencomparing the two policies, the effects of education subsidies are relatively small, whileaccounting for child-labor regulations is crucial.

The results show that accounting for policy changes is important for understanding theexperience of different countries during the transition from stagnation to growth. Thisfinding does not rule out that other factors could play an important role as well. For ex-ample, the model abstracts from changes in mortality rates, even though declining mor-tality is an equally important aspect of the demographic transition as declining fertility.As a simple extension to the model, changes in child mortality can be incorporated byintroducing a probability of survival until adulthood. In the calibrated model, for smallincreases in child mortality the number of births increases, but the number of surviv-ing children declines. Feeding realistic survival rates into the model makes it possible tomatch actual fertility numbers more closely, but the effects of policy changes are still thesame and dominate the effects of changing child mortality.19 In order to incorporate adultmortality, the framework would have to be extended to allow for more than two periodsof life.

Another question is how the stylized policies in the model are implemented in the realworld. Child-labor regulations are subject to an obvious enforcement problem, which issevere given that much child labor takes place in agriculture, where direct supervision isdifficult. A number of studies have found that official child-labor restrictions do little toconstrain child labor. In practice, compulsory schooling laws often appear to be a more

18The role of child labor in determining fertility has been analyzed by Rosenzweig and Evenson (1980).In line with the results here, they find that the economic contributions of children through child labor arean important determinant for fertility rates.

19Within the dynastic utility framework, changes in child mortality rates have only minor effects onfertility even if stochastic survival and sequential fertility choice are taken into account, see Doepke (2002).

28

effective constraint on child labor than direct regulation.20 From an administrative pointof view, it is easier to check that children show up in a given classroom at a certain time,as opposed to ascertaining that they do not engage in illegal work at any time and anyplace. Therefore, even though the policies can be used separately in the model, they arebest thought of as a joint policy in practice.

In the discussion of the transition from stagnation to growth I concentrated on cross-country differences in the fertility decline. An important question that is not addressedin this paper is why the take-off occurs at different times in different countries. The gov-ernment policies that are discussed in this paper have the potential to cause a delay inthe take-off of about 10 to 20 years, which is clearly too little to explain the observed dif-ferences. In order to make progress along this line, it will be necessary to move beyondthe assumption of exogenous productivity growth and introduce a theory of technologi-cal progress. The results in this paper do not depend in any way on the assumption thatproductivity growth is exogenous. As a first step beyond exogenous growth, it is possibleto link the rate of technological progress to the number of skilled people in the economy.Such a model can explain why the rate of economic growth increased during the indus-trial revolution, instead of jumping to the growth-regime level right away. The effects ofgovernment policies on growth would be amplified.

Going beyond the specific application to fertility decline, this paper contributes to anemerging literature which puts the spotlight on the role of policy reforms in the courseof development. Every now industrialized country has introduced child labor and publiceducation laws at a certain stage of development. The same is true about other policiessuch as social insurance programs. While the research in this paper suggests that policyreforms can have large effects on fertility decisions, an intriguing possibility for futureresearch would be to examine the reverse channel. If family size is dependent on educa-tional policies, conversely voters’ preferences over such policies should depend at leastpartly on their earlier fertility decisions. This suggests the possibility of a two-way in-teraction between the demographic changes and the political reforms that characterizeeconomies during the transition from stagnation to growth.

20For example, using U.S. data, Margo and Finegan (1996) find that child-labor restrictions are especiallyeffective if combined with compulsory schooling laws.

29

References

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Barro, Robert J., and Jong-Wha Lee. 1993. International Comparisons of EducationalAttainment. Journal of Monetary Economics 32:363–394.

———. 1997. Schooling Quality in a Cross Section of Countries. NBER Working Paper6198.

Becker, Gary S., and Robert J. Barro. 1988. A Reformulation of the Economic Theory ofFertility. Quarterly Journal of Economics 53(1):1–25.

Bloom, David E., and Jeffrey G. Williamson. 1998. Demographic Transitions and Eco-nomic Miracles in Emerging Asia. World Bank Economic Review 12(3):419–55.

de la Croix, David, and Matthias Doepke. 2003. Inequality and Growth: Why DifferentialFertility Matters. American Economic Review 93(4):1091–1113.

Chesnais, Jean Claude. 1992. The Demographic Transition. Oxford: Oxford UniversityPress.

Dahan, Momi, and Daniel Tsiddon. 1998. Demographic Transition, Income Distribution,and Economic Growth. Journal of Economic Growth 3:29–52.

Deane, Phyllis, and W.A. Cole. 1969. British Economic Growth 1688–1959. Cambridge:Cambridge University Press.

Deininger, Klaus, and Lyn Squire. 1996. Measuring Income Inequality: A New Database.Development Discussion Paper No. 537, Harvard Institute for International Develop-ment.

Doepke, Matthias. 2002. Child Mortality and Fertility Decline: Does the Barro-BeckerModel Fit the Facts? UCLA Department of Economics Working Paper No. 824.

Fernandez-Villaverde, Jesus. 2001. Was Malthus Right? Economic Growth and Popula-tion Dynamics. Mimeo, University of Pennsylvania.

Galor, Oded, and David N. Weil. 1996. The Gender Gap, Fertility, and Growth. AmericanEconomic Review 86(3):375–87.

———. 2000. Population, Technology, and Growth: From Malthusian Stagnation to theDemographic Transition and Beyond. American Economic Review 90(4):806–828.

Greenwood, Jeremy, and Ananth Seshadri. 2002. The U.S. Demographic Transition. Amer-ican Economic Review Papers and Proceedings 92(2):153–159.

Hansen, Gary D., and Edward C. Prescott. 2002. Malthus to Solow. American EconomicReview 92(4):1205–1217.

Haveman, Robert, and Barbara Wolfe. 1995. The Determinants of Children’s Attainments:

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A Review of Methods and Findings. Journal of Economic Literature 33:1829-1878.

Horrell, Sara, and Jane Humphries. 1995. The Exploitation of Little Children: Child Laborand the Family Economy in the Industrial Revolution. Explorations in Economic History32:485-516.

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Jones, Charles I. 2001. Was an Industrial Revolution Inevitable? Economic Growth overthe Very Long Run. Advances in Macroeconomics 1(2): Article 1.

Jones, Elise F. 1982. Socio-Economic Differentials in Achieved Fertility. World Fertility SurveyComparative Studies No. 21. Voorburg, Netherlands: International Statistical Institute.

Kogel, Tomas, and Alexia Prskawetz. 2001. Agricultural Productivity Growth and theEscape from the Malthusian Trap. Journal of Economic Growth 6:337–57.

Knowles, John. 1999. Can Parental Decisions Explain U.S. Income Inequality? Workingpaper, University of Pennsylvania.

Kremer, Michael, and Daniel Chen. 1999. Income Distribution Dynamics with Endoge-nous Fertility. American Economic Review 89(2): 155–160.

Kuznets, Simon. 1955. Economic Growth and Income Inequality. American EconomicReview 45:1–28.

Lagerlof, Nils-Petter. 2002. Gender Equality and Long-Run Growth. Forthcoming, Jour-nal of Economic Growth.

Maddison, Angus. 1991. Dynamic Forces in Capitalist Development: A Long-Run ComparativeView. Oxford: Oxford University Press.

Margo, Robert A. and Finegan, T. Aldrich. 1996. Compulsory Schooling Legislationand School Attendance in Turn-of-the-Century America: A “Natural Experiment” Ap-proach. Economics Letters 53:103–10.

Moav, Omer. 2001. Cheap Children and the Persistence of Poverty. CEPR DiscussionPaper No. 3059.

Morand, Olivier F. 1999. Endogenous Fertility, Income Distribution, and Growth. Journalof Economic Growth 4(3):331–49.

Nardinelli, Clark. 1980. Child Labor and the Factory Acts. Journal of Economic History40(4):739–55.

Ngai, Liwa Rachel. 2000. Barriers and the Transition to Modern Growth. Mimeo, Univer-sity of Pennsylvania.

Park, Young-Bum, David R. Ross, and Richard H. Sabot. 1996. Educational Expansionand the Inequality of Pay in Brazil and Korea. In: Opportunity Forgone: Education inBrazil. Nancy Birdsall and Richard H. Sabot, edts. Inter-American Development Bank:Washington, D.C.

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A Definition of an Equilibrium

This section introduces the remaining equilibrium conditions, starting with the determi-nation of labor supply. Skilled adults distribute their time between working, raising andteaching their own children, and teaching children of unskilled parents. Therefore thetotal supply of skilled labor LS is given by:

LS(x) =[1 − (φ + φS) λS→S(x) nS(S, x) − φ λS→U (x) nU (S, x)] NS

− φS λU→S(x) nS(U, x) NU . (16)

Unskilled labor LU is supplied by unskilled adults and by children who do not go toschool:

LU (x) =[1 − φ λU→S(x) nS(U, x) − φ λU→U (x) nU (U, x)] NU

+ φU (x) [λS→U (x) nU (S, x) NS + λU→U (x) nU (U, x) NU ]. (17)

In equilibrium, labor supply has to equal labor demand for each type of labor. I assumethat skilled adults can perform both the skilled and the unskilled work, while unskilledadults can do unskilled work only. Under this assumption, the skilled wage cannot fallbelow the unskilled wage:

wS(x) ≥ wU (x). (18)

The market-clearing conditions for the labor market are:

LFS(x) + LIS(x) ≤LS(x), = if wS(x) > wU (x), (19)LFU (x) + LIU (x) =LU (x) + [LS(x) − LFS(x) − LIS(x)] . (20)

32

The flat tax τ which finances the education subsidy has to be chosen to observe budgetbalance. The correct tax rate is given by dividing the total expenditure on schooling sub-sidies by total wage income:

τ (x) =δ(x) φS N ′

S wS(x)LS(x) wS(x) + LU (x) wU (x) + φS N ′

S wS(x). (21)

Here N ′S is the total number of skilled children. Notice that for the computation of total

labor income we have to add the income of the teachers to the wage of the usual workers.Teachers receive wages for their work and are taxed like all other adults in the modeleconomy.

The final equilibrium condition is the law of motion for population. Since I abstract fromchild mortality, the number of adults of a given type tomorrow is given by the number ofchildren of that type today:

N ′S =λS→S(x) nS(S, x) NS + λU→S(x) nS(U, x) NU , (22)

N ′U =λS→U (x) nU (S, x) NS + λU→U (x) nU (U, x) NU . (23)

We now have all the ingredients at hand that are needed to define an equilibrium.

Definition 1 (Recursive Competitive Equilibrium) Given a government policy {φU , δ}, arecursive competitive equilibrium consists of a tax function τ , value functions VS and VU , laborsupply functions LS and LU , labor demand functions LFS , LFU , LIS, and LIU , wage functionswS and wU , mobility functions λS→S, λS→U , λU→S, and λU→U , all mapping X into R+, policyfunctions nS and nU mapping {S, U} × X into R+, and a law of motion g mapping X into itself,such that:

(i) The value functions satisfy the following functional equation for i ∈ {S, U}:

Vi(x) = maxc,nS,nU≥0

{cσ + β(nS + nU )−ε

[nSVS(x′) + nUVU (x′)

]}. (24)

subject to the budget constraint (9) and the law of motion x′ = g(x).

(ii) For i, j ∈ {S, U}, if λi→j(x) > 0, nj(i, x) attains the maximum in (24).

(iii) The tax function τ satisfies the government budget constraint (21).

(iv) The wages wS and wU and labor demand LFS , LFU , LIS, and LIU satisfy (5)–(8) and (18).

(v) Labor supply LS and LU satisfies (16) and (17).

(vi) Labor supply LS and LU and labor demand LFS, LFU , LIS , and LIU satisfy (19) and (20).

(vii) The mobility functions λS→S, λS→U , λU→S, and λU→U satisfy (10).

33

(viii) The law of motion g for the state variable x is given by (4), (22), and (23).

The equilibrium conditions do not include a market-clearing constraint for the goods mar-ket, because it holds automatically by Walras’ Law. In condition (ii) above, it is under-stood that parents choose only one type of children. In other words, saying that nS(S, x)attains the maximum in (24) means that {nS = nS(S, x), nU = 0} and the consumption cthat results from the budget constraint maximize utility. Maximization is only requiredif a positive number of parents choose the type of children in question. Condition (iii)is the government budget constraint. Condition (iv) requires that wages equal marginalproducts and that the skilled wage does not fall below the unskilled wage, condition (v)links labor supply to population and education time, condition (vi) is the market-clearingcondition for the labor market, condition (vii) requires that for each type of adult the frac-tions having skilled and unskilled children sum to one, and condition (viii) defines thelaw of motion.

B Proofs for all Propositions

Proof of Proposition 1 The profit-maximization problem of a firm in the industrial sectoris given by:

maxlS ,lU

{AI (lS)1−α (lU )α − wS(x) lS − wU (x) lU

}. (25)

The first-order condition for a maximum with respect to lU gives:

(lS)1−α =wU (x)(lU )1−α

AIα.

Plugging this expression back into (25) yields a formulation of the profit maximizationproblem as a function of unskilled labor only:

maxlU

{wU (x)

αlU − wS(x)

(wU (x)AIα

) 11−α

lU − wU (x) lU

}. (26)

Since this expression is linear in lU , production in the industrial sector will be profitableonly if we have:

wU (x)α

− wS(x)(

wU (x)AIα

) 11−α

− wU (x) ≥ 0,

which can be rearranged to get:

wS(x)1−α wU (x)α ≤ AI (1 − α)1−α αα,

which is (11).

34

The first-order necessary conditions for a maximum of the profit-maximization problemof a firm in agriculture are given by the wage conditions (5) and (6):

wS(x) =AFθS

θS + θU

LθUFU

L1−θSFS

Z1−θS−θU , (27)

wU (x) =AFθU

θS + θU

LθSFS

L1−θUFU

Z1−θS−θU . (28)

Since the objective function is concave, the first-order conditions are also sufficient for amaximum. It is therefore sufficient to show that for any wS(x), wU (x) > 0 we can findvalues for skilled and unskilled labor supply LFS and LFU such that (27) and (28) aresatisfied. The required values are given by:

LFS =(

AF

θS + θU

) 11−θS−θU

(θS

wS(x)

) 1−θS1−θS−θU

(θU

wU (x)

) θU1−θS−θU

Z

and:

LFU =(

AF

θS + θU

) 11−θS−θU

(θS

wS(x)

) θS1−θS−θU

(θU

wU (x)

) 1−θU1−θS−θU

Z,

which are positive for any positive wages wS(x) and wU (x). �

Proof of Proposition 2 We consider the following maximization problem:

maxE≥0,0≤f≤0

{(w − E)σ + βE1−ε(f/pS + (1 − f )/pU )−ε[fVS/pS + (1 − f )VU/pU ]

}.

By assumption, the parameters β, σ and ε are all strictly bigger than zero and strictlysmaller than one. It is also assumed w.l.o.g. that 0 < pU < pS and VS, VU ≥ 0 hold. Wewant to show that there are no interior solutions in f , i.e., if there is a solution to themaximization problem above, the optimal f is either zero or one.

To show that there are no interior solutions, assume that we have already determinedthe optimal E. Given this E and the fact that the function to be maximized is twicecontinuously differentiable in f , if there were an interior solution, the optimal f wouldhave to satisfy first- and second-order conditions for a maximum. I solve for the uniquef which solves the first-order condition, and show that this f does not satisfy the second-order condition. This proves that there are only corner solutions.

I will name the maximand U(·). The first and second derivatives of U with respect to f

35

are:

∂U

∂f= βE1−ε

[−ε

(1pS

− 1pU

) (f

pS+

1 − f

pU

)−ε−1 (fVS

pS+

(1 − f )VU

pU

)

+(

f

pS+

1 − f

pU

)−ε (VS

pS− VU

pU

)],

∂2U

∂f 2 = βE1−εε

(1pS

− 1pU

) (f

pS+

1 − f

pU

)−ε−1

[(1 + ε)

(1pS

− 1pU

) (f

pS+

1 − f

pU

)−1 (fVS

pS+

(1 − f )VU

pU

)− 2

(VU

pU− VS

pS

) ].

In the first derivative, for 0 ≤ f ≤ 1, the first term within the outer brackets is positive.For an interior solution to be possible, it has to be the case that VS/pS < VU/pU , becauseotherwise the second term is also positive and the first-order condition cannot be satisfied.Therefore if VS/pS ≥ VU/pU , we are done. For the case that VS/pS < VU/pU , setting thefirst derivative equal to zero and solving for f yields:

f =ε(

VU

pS− VU

pU

)−

(VS

pS− VU

pU

)(1 − ε)pU

(VS

pS− VU

pU

)(1

pS− 1

pU

) .

I will now plug this value for f into the second derivative to verify that the second deriva-tive is positive, so that the critical point is not a maximum. The second derivative ispositive if the following inequality holds:

(1 + ε)(

1pS

− 1pU

) (f

pS+

1 − f

pU

)−1 (fVS

pS+

(1 − f )VU

pU

)− 2

(VS

pS− VU

pU

)< 0.

Plugging in our value for f yields after some algebra:

(1 + ε)1pU

(VU

pS

− VU

pU

)<

1pU

[ε

(VU

pS

− VU

pU

)−

(VS

pS

− VU

pU

)]+ 2

1pU

(VS

pS

− VU

pU

),

1pU

(VU

pS

− VU

pU

)<

1pU

(VS

pS

− VU

pU

),

VU < VS,

Thus if VS > VU the second-order condition for a maximum is not satisfied, and thereforethere is no interior maximum. Conversely, if VS were smaller than VU , there would beonly unskilled children for sure, since they are cheaper to educate. Thus once again weobtain a corner solution. Thus in every case there are only corner solutions. families have

36

either unskilled or skilled children, but they do not mix. �

Proof of Proposition 3 It is helpful to consider the formulation of the problem in whichadults choose the total education cost E, so that the number of children equals E/pi foradults who choose to have children of type i. The maximization problem in this formula-tion is:

max0≤E≤w/pi

{(w − E)σ + β(E/pi)1−εVi

}. (29)

This can also be written as:

maxE≥0

{(w − E)σ + β(E)1−ε Vi

(pi)1−ε

}. (30)

Since the costs and utilities of children enter only in the last term, an adult is indifferentbetween skilled and unskilled children if and only if:

VS

(pS)1−ε=

VU

(pU )1−ε, (31)

Notice that this condition does not depend on the wage of the adult. Also, if condition (31)is satisfied, adults face the same maximization problem regardless whether they decidefor unskilled or skilled children. This implies that the optimal total education cost E doesnot depend on the type of the children. The higher cost of having skilled children will beexactly made up by a lower number of children. �

Proof of Proposition 4 Since we assume wS > wU , i.e., the skilled wage strictly exceedsthe unskilled wage, the time of skilled parents is more expensive. We then have:

φwS + φSwS + ρ

φwS − φUwU + ρ<

φwU + φSwS + ρ

φwU − φUwU + ρ,

the ratio of the total cost of skilled child to that an unskilled child is lower for skilled par-ents than for unskilled parents. In other words, skilled children are relatively cheaper forskilled parents, and unskilled children are relatively cheaper for unskilled parents. Sincein equilibrium there must be a positive number of children of each type, at least someskilled parents must have skilled children, and at least some unskilled parents must haveunskilled children. Moreover, it follows from Proposition 3 that both types of parents canbe indifferent between both types of children only if the price ratio between skilled andunskilled children is the same. This cannot be the case since the wage and therefore thetime cost differs, thus just one type of parent can be indifferent. �

37

C Policies in Brazil, Korea, and England

Korea and Brazil are polar cases in terms educational and child-labor policies. Korea in-stituted a system of free, compulsory education from the ages of 6 to 12 in 1949, rightafter independence. After the Korean war the government instituted a ‘Compulsory Ed-ucation Accomplishment Plan,” and by 1959 the primary enrollment rate reached 96%.Strictly enforced compulsory education served as an effective constraint on child labor, inaddition to direct child-labor restrictions. According to the International Labor Organiza-tion (ILO), in 1960 only 1.1% of the children from zero to fifteen years were economicallyactive, and by 1985 only .3% of the children between ages ten and fourteen participated inthe labor market. By the letter of the law, free and compulsory education was introducedin Brazil already in 1930. In practice, however, primary schooling was not available inmany rural areas, and to the present day the available schools are often of poor quality.In 1965 the primary enrollment rate was still below 50% in rural areas. The neglect ofprimary education in Brazil can also be gauged from education finances. In 1960, lessthan 10% of public spending on education was directed to the primary sector, while thecorresponding figure was about 70% for Korea. Child-labor regulation is less restrictivein Brazil in Korea. While Korea signed on to an ILO convention that rules out child la-bor under the age of 14, Brazil did not. The minimum age for employment in Brazil isnow 12, and even this limit is not always enforced. In 1985 18.7% of the children betweenages ten and fourteen participated in the labor market. Child labor is even more preva-lent among male children, with 25.3% listed as economically active in 1987. The differentpolicies in Brazil and Korea are also reflected in educational outcomes. In both countriesaverage years of schooling increase over time, but the increase is much faster in Korea.In 1960, the difference in average years of schooling between Korea and Brazil was lessthan a year. By 1985, the difference increased to more than four years, so that Koreanson average have more than twice as much schooling as Brazilians. Another measure ofeducational success is given by adult illiteracy rates. In 1960, 39% of adult Brazilians wereilliterate, compared to 29% of adult Koreans. In 1995, in Brazil illiteracy is still at 16.7%,while Koreans are almost completely literate with an illiteracy rate of only 2%. In Brazililliteracy is high even among groups who went to school only recently. In 1991, illiteracywas 12.1% for the age group from 15 to 19 years.

In England, education was not widespread at the beginning of the Industrial Revolution.In 1780, only about 50% of brides and grooms were able to sign their name. Educationexpanded only slowly throughout most of the nineteenth century. Public education alsostarted to expand in the nineteenth century, but there were large regional differences andno universal access to affordable education. The situation changed drastically with theVictorian education reforms. The Forster Education Act of 1870 placed primary educationunder public control. In 1880 compulsory schooling was introduced, and starting in 1891primary education was free. While schooling quality is hard to measure, literacy datasuggests that the reforms were successful. While in 1880 about 15% of grooms and 20%of brides were unable to sign their name, these numbers decreased to less than 2% until

38

1910. The first child-labor restrictions were put in place with the Factory Act of 1833. Onlya small set of industries was affected, however, and Nardinelli (1980) concludes that theimpact on child labor was small. The Factory Acts were amended in 1844 and 1874, whenthe minimum age for child laborers was raised to 10. At that time the restrictions becameuniversal, instead of being limited to certain industries. Compulsory schooling laws alsohad an effect on child labor. As a result, the incidence of child labor was decreasing latein the nineteenth century. Activity rates for children aged from ten to fourteen reached apeak 1861, when 29% of all children in that age group were economically active. In 1871,the number was still at 26%, but then it fell to about 20% in 1881 and 1891, 17% in 1901,and 14% in 1911.

D Notes on Computation

Within the Malthusian regime and the growth regime, the model can be computed viastandard value function iteration on a discretized state space. The initial guess for thevalue functions is computed by setting the number of children to zero, so that utilitystems from consumption only. During the iterations, for a given state x, the algorithmfinds a state x′ in the next period such that the resulting fertility decisions of adults areconsistent with state x′. After the iterations converge, the equilibrium law of motion canbe used to compute steady-state values. Alternatively, the steady state can be computedby solving a system of equations that defines the steady state.

In principle it is possible to compute the entire model, encompassing the Malthusianregime, the transition, and the growth regime, with the same method described above.However, since the state vector is four-dimensional, computations would either take verylong or would be imprecise. Therefore I use a shooting algorithm that directly computesthe equilibrium path over T periods from any starting value x0 for the state vector. Thecomputations are started a number of periods before the industrial technology becomescompetitive. The number of periods T to be computed has to be chosen sufficiently largesuch that the economy is close to the balanced growth path at T . Whether this is the casecan be checked by computing the growth regime as described above before computingthe transition. The algorithm is not guaranteed to converge, but it works well in practice.

39

1000 1500 2000 2500 3000 3500 40002

2.5

3

3.5

4

4.5

5

5.5

6

6.5

TFR

GDP per capita

Brazil EnglandKorea

The graph covers approximately the periods 1820-1914 for Eng-land, 1945 to 1980 for Brazil, and 1960 to 1985 for Korea.

Figure 1: Fertility Decline from Peak Relative to GDP per capita in Brazil, Korea, andEngland

1900 1950 2000 20500

2

4

6

8

GD

P/C

apita

No Reform − "Brazil"

1900 1950 2000 2050

2

3

4

TFR

No Reform − "Brazil"

1750 1800 1850 19000

2

4

6

8

GD

P/C

apita

Delayed Reform − "England"

1750 1800 1850 1900

2

3

4

TFR

Delayed Reform − "England"1900 1950 2000 20500

2

4

6

8

GD

P/C

apita

Immediate Reform − "Korea"

1900 1950 2000 2050

2

3

4

TFR

Immediate Reform − "Korea"

Figure 2: Outcome with Different Policies (Dotted Lines are Brazilian Outcomes as aBenchmark)

1900 1950 2000 20500

0.2

0.4

0.6

Gin

iNo Reform − "Brazil"

1900 1950 2000 20500

1

2

3

4

TFR

No Reform − "Brazil"

Skilled Unskilled

1750 1800 1850 19000

0.2

0.4

0.6

Gin

i

Delayed Reform − "England"

1750 1800 1850 19000

1

2

3

4T

FRDelayed Reform − "England"

Skilled Unskilled

1900 1950 2000 20500

0.2

0.4

0.6

Gin

i

Immediate Reform − "Korea"

1900 1950 2000 20500

1

2

3

4

TFR

Immediate Reform − "Korea"

Skilled Unskilled

Figure 3: Gini Coefficients and Fertility Differentials

1900 1950 2000 20500

2

4

6

8

GD

P/C

apita

Child−Labor Restriction Only

1900 1950 2000 2050

2

3

4

TFR

Child−Labor Restriction Only

1900 1950 2000 20500

2

4

6

8

GD

P/C

apita

Education Subsidy Only

1900 1950 2000 2050

2

3

4

TFR

Education Subsidy Only

Figure 4: Outcome with Isolated Policies